Graph Saturation in Color Michael Ferrara Jaehoon Kim Elyse Yeager* yeager2@illinois.edu UAA Mathematics Colloquium February 2015 Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 1 / 22 Outline 1 Introduction to Graphs 2 Graph Saturation 3 Ramsey Theory 4 Colored Graph Saturation Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 2 / 22 What Do You Know About Graphs? Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 3 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics vertex |G | : number of vertices in G Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics vertex |G | : number of vertices in G Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics edge vertex |G | : number of vertices in G kG k : number of edges in G Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Graph Basics Subgraph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 4 / 22 Common Graphs Path: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Matching: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Matching: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Matching: Clique: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Matching: Clique: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Matching: Clique: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Common Graphs Path: Matching: Clique: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 5 / 22 Outline 1 Introduction to Graphs 2 Graph Saturation 3 Ramsey Theory 4 Colored Graph Saturation Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 6 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. This graph is triangle saturated Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. This graph is triangle saturated Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. This graph is triangle saturated Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Definition A graph G is H-saturated if: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Definition A graph G is H-saturated if: 1 H is not a subgraph of G Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Goal: Avoid a forbidden subgraph, but add as many edges as possible. Example: Let the triangle be forbidden. Definition A graph G is H-saturated if: 1 H is not a subgraph of G and 2 If we add any edge to G , H is a subgraph of the resulting graph. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 7 / 22 Saturation Number Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number 7 edges Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number 7 edges 8 edges Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number 7 edges 8 edges 5 edges Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number 7 edges 8 edges 5 edges Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number 7 edges 8 edges 5 edges Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Given the above examples: sat(6; triangle) Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Saturation Number 7 edges 8 edges 5 edges Definition: (Erdős-Hajnal-Moon) The saturation number of a graph H and a number n is the minimum number of edges in a graph on n vertices that is H saturated. sat(n; H) := min{kG k : |G | = n and G is H saturated} Given the above examples: sat(6; triangle) ≤ 5 Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 8 / 22 Outline 1 Introduction to Graphs 2 Graph Saturation 3 Ramsey Theory 4 Colored Graph Saturation Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 9 / 22 Edge Coloring Definition: An edge coloring of a graph is an assignment of a color to each edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 10 / 22 Edge Coloring Definition: An edge coloring of a graph is an assignment of a color to each edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 10 / 22 Edge Coloring Definition: An edge coloring of a graph is an assignment of a color to each edge. Colors: red, blue. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 10 / 22 Edge Coloring Definition: An edge coloring of a graph is an assignment of a color to each edge. Colors: red, blue. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 10 / 22 Edge Coloring Definition: An edge coloring of a graph is an assignment of a color to each edge. Colors: red, blue. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 10 / 22 Edge Coloring Definition: An edge coloring of a graph is an assignment of a color to each edge. Definition: A graph is monochromatic if all its edges are assigned the same color. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 10 / 22 Ramsey Problems Goal: An edge-coloring with no forbidden monochromatic subgraph. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 11 / 22 Ramsey Problems Goal: An edge-coloring with no forbidden monochromatic subgraph. Example: Color red and blue; forbid red triangle and blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 11 / 22 Ramsey Problems Goal: An edge-coloring with no forbidden monochromatic subgraph. Example: Color red and blue; forbid red triangle and blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 11 / 22 Ramsey Problems Goal: An edge-coloring with no forbidden monochromatic subgraph. Example: Color red and blue; forbid red triangle and blue triangle. good coloring Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 11 / 22 Ramsey Problems Goal: An edge-coloring with no forbidden monochromatic subgraph. Example: Color red and blue; forbid red triangle and blue triangle. good coloring Ferrara-Kim-Yeager (UCD, UIUC) bad coloring Graph Saturation in Color Feb 2015 11 / 22 Ramsey Problems Goal: An edge-coloring with no forbidden monochromatic subgraph. Example: Color red and blue; forbid red triangle and blue triangle. good coloring Ferrara-Kim-Yeager (UCD, UIUC) bad coloring Graph Saturation in Color bad coloring Feb 2015 11 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Example: Any red-blue coloring of the edges of K6 contains a monochromatic red triangle or a monochromatic blue triangle. no good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Ramsey’s Theorem Given any collection of forbidden subgraphs, and any sufficiently large clique, no good coloring exists. The lowest “sufficiently large” number for a given collection of subgraphs is called the Ramsey Number. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Ramsey’s Theorem Given any collection of forbidden subgraphs, and any sufficiently large clique, no good coloring exists. The lowest “sufficiently large” number for a given collection of subgraphs is called the Ramsey Number. R(3, 3) = 6 (two triangles) Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Ramsey’s Theorem Given any collection of forbidden subgraphs, and any sufficiently large clique, no good coloring exists. The lowest “sufficiently large” number for a given collection of subgraphs is called the Ramsey Number. R(3, 3) = 6 (two triangles) R(4, 4) = 18 Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Ramsey’s Theorem Given any collection of forbidden subgraphs, and any sufficiently large clique, no good coloring exists. The lowest “sufficiently large” number for a given collection of subgraphs is called the Ramsey Number. R(3, 3) = 6 (two triangles) R(4, 4) = 18 43 ≤ R(5, 5) ≤ 49 Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Ramsey Problems Note: Sometimes a good coloring doesn’t exist! Ramsey’s Theorem Given any collection of forbidden subgraphs, and any sufficiently large clique, no good coloring exists. Erdős-Szekeres: Given N, any sufficiently large collection of points in general position contains a subset forming the vertices of a convex N-gon Van der Waerden: Any coloring of the natural numbers contains arbitrarily long monochromatic arithmetic sequences. Green-Tao: The sequence of prime numbers contains arbitrarily long arithmetic progressions. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 12 / 22 Outline 1 Introduction to Graphs 2 Graph Saturation 3 Ramsey Theory 4 Colored Graph Saturation Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 13 / 22 Marrying The Two Recall: A graph G is H-saturated if G contains no H subgraph, but adding any edge to G creates an H subgraph. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 14 / 22 Marrying The Two Recall: A graph G is H-saturated if G contains no H subgraph, but adding any edge to G creates an H subgraph. Ramsey version of saturation: Given forbidden graphs H1 , . . . , Hk (in colors 1, . . . , k respectively), we say a graph G is (H1 , . . . , Hk )-saturated if a good edge-coloring of G exists (using colors 1, . . . , k), but if we add any edge to G , all colorings are bad. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 14 / 22 Marrying The Two Example: the graph below is Ferrara-Kim-Yeager (UCD, UIUC) , Graph Saturation in Color -saturated. Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 Marrying The Two Example: the graph below is , -saturated. First: show a good coloring exists. Second: show no good coloring exists if we add any edge. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 15 / 22 rsat(n; H1 , . . . , Hk ) Again, we are interested in (H1 , . . . , Hk )-saturated graphs with as few edges as possible. Definition: For a number n and forbidden subgraphs H1 , . . . , Hk , we define rsat(n; H1 , . . . , Hk ) to be the minimum number of edges over all n-vertex graphs that are (H1 , . . . , Hk )-saturated. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 16 / 22 rsat(n; H1 , . . . , Hk ) Again, we are interested in (H1 , . . . , Hk )-saturated graphs with as few edges as possible. Definition: For a number n and forbidden subgraphs H1 , . . . , Hk , we define rsat(n; H1 , . . . , Hk ) to be the minimum number of edges over all n-vertex graphs that are (H1 , . . . , Hk )-saturated. Previous Example: is Ferrara-Kim-Yeager (UCD, UIUC) , -saturated Graph Saturation in Color Feb 2015 16 / 22 rsat(n; H1 , . . . , Hk ) Again, we are interested in (H1 , . . . , Hk )-saturated graphs with as few edges as possible. Definition: For a number n and forbidden subgraphs H1 , . . . , Hk , we define rsat(n; H1 , . . . , Hk ) to be the minimum number of edges over all n-vertex graphs that are (H1 , . . . , Hk )-saturated. Previous Example: is Ferrara-Kim-Yeager (UCD, UIUC) , -saturated, so rsat 5; Graph Saturation in Color , ≤ 6. Feb 2015 16 / 22 Forbidding Cliques: Ramsey Numbers Definition: Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 17 / 22 Forbidding Cliques: Ramsey Numbers Definition: Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists. Example: R(3, 3) = 6 Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 17 / 22 Forbidding Cliques: Ramsey Numbers Definition: Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists. Example: R(3, 3) = 6 no good coloring of K6 exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 17 / 22 Forbidding Cliques: Ramsey Numbers Definition: Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists. Example: R(3, 3) = 6 no good coloring of K6 exists Ferrara-Kim-Yeager (UCD, UIUC) a good coloring of K5 exists Graph Saturation in Color Feb 2015 17 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 Ferrara-Kim-Yeager (UCD, UIUC) n − (R − 2) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists If we add any edge, NO good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists If we add any edge, NO good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) A good coloring exists If we add any edge, NO good coloring exists Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Forbidding Cliques: Hanson-Toft Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated. KR−2 n − (R − 2) Hanson-Toft Conjecture: The construction above has the fewest possible edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 18 / 22 Hanson-Toft Hanson-Toft Conjecture sat(n; Rmin (Kk1 , . . . , Kkt )) = r −2 2 Ferrara-Kim-Yeager (UCD, UIUC) n 2 n<r + (r − 2)(n − r + 2) n ≥ r Graph Saturation in Color Feb 2015 19 / 22 Hanson-Toft Hanson-Toft Conjecture sat(n; Rmin (Kk1 , . . . , Kkt )) = r −2 2 n 2 n<r + (r − 2)(n − r + 2) n ≥ r Chen, Ferrara, Gould, Magnant, Schmitt; 2011 sat(n; Rmin (K3 , K3 )) = Ferrara-Kim-Yeager (UCD, UIUC) n 2 n<6=r 4n − 10 n ≥ 56 Graph Saturation in Color Feb 2015 19 / 22 Hanson-Toft Hanson-Toft Conjecture sat(n; Rmin (Kk1 , . . . , Kkt )) = r −2 2 n 2 n<r + (r − 2)(n − r + 2) n ≥ r Chen, Ferrara, Gould, Magnant, Schmitt; 2011 sat(n; Rmin (K3 , K3 )) = Ferrara-Kim-Yeager (UCD, UIUC) n 2 n<6=r 4n − 10 n ≥ 56 Graph Saturation in Color Feb 2015 19 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Matching Saturation: Mader, 1973 Forbidden: Graph: No forbidden subgraph Ferrara-Kim-Yeager (UCD, UIUC) Adding any edge creates forbidden subgraph Graph Saturation in Color Feb 2015 20 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Example: Forbidden matching on 10 edges. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Forbidding Matchings: Ferrara-Kim-Y Colored Example: Forbidden Ferrara-Kim-Yeager (UCD, UIUC) , , Graph Saturation in Color , . Feb 2015 21 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs Ferrara-Kim-Yeager (UCD, UIUC) , . Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs Ferrara-Kim-Yeager (UCD, UIUC) , . Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy take red subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy take red subgraph Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy take red subgraph This (uncolored!) subgraph is saturated. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy take red subgraph This (uncolored!) subgraph is saturated. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy take red subgraph This (uncolored!) subgraph is saturated. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Iterated Recoloring: Ferrara-Kim-Yeager Goal: Use results from (uncolored) saturation in the Ramsey version. Example: Forbidden graphs , . good coloring make red-heavy take red subgraph This (uncolored!) subgraph is saturated. Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 22 / 22 Thanks for Listening! Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 23 / 22